Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 8: Problem 68
Change each equation to rectangular coordinates and then graph.Graph \(r_{1}=2+2 \cos \theta\) and \(r_{2}=2-2 \cos \theta\) and name three points they have in common.
Short Answer
Expert verified
The circles intersect at points \((0, 2)\), \((0, -2)\), and \((0, 0)\).
Step by step solution
01
Understanding the Polar Coordinates
The given equations are in polar coordinates: \(r_1 = 2 + 2\cos\theta\) and \(r_2 = 2 - 2\cos\theta\). These are known as polar curves. To convert them into rectangular coordinates (\(x,y\)), we need to use the relations \(x = r\cos\theta\) and \(y = r\sin\theta\).
02
Conversion of the Equation \(r_1 = 2 + 2\cos\theta\) to Rectangular Coordinates
For the first equation, we use the identity \(r = \sqrt{x^2 + y^2}\) and \(x = r\cos\theta\), so \(r_1 = 2 + 2\frac{x}{r_1}\). This becomes \(r_1^2 = (2 + \frac{2x}{r_1})^2\). By multiplying through by \(r_1^2\) and simplifying, we get \((x - 2)^2 + y^2 = 4\), which represents a circle centered at \((2, 0)\) with radius 2.
03
Conversion of the Equation \(r_2 = 2 - 2\cos\theta\) to Rectangular Coordinates
Similarly, using the same approach, \(r_2 = 2 - 2\frac{x}{r_2}\) becomes \((x + 2)^2 + y^2 = 4\) after simplification. This represents a circle centered at \((-2, 0)\) with radius 2.
04
Finding Intersection Points
Both equations represent circles, and to find their intersection points, set \((x - 2)^2 + y^2 = (x + 2)^2 + y^2\). Simplify to find \(x = 0\). Substitute \(x = 0\) back into either circle's equation, \(y^2 = 4 - 0\). Thus, \(y = \pm 2\). The common points are \((0, 2)\) and \((0, -2)\). The polar curves also intersect at \(\theta = \pi/2\), giving the point \((0, 0)\) as another intersection due to their symmetry.
05
Graphing and Finalizing the Solution
Graph the circles centered at \((2, 0)\) and \((-2, 0)\) each with radius 2. The intersection points \((0, 2)\), \((0, -2)\), and \((0, 0)\) are visible where the circles intersect. These are the three points the circles have in common.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Polar Coordinates
Polar coordinates are a way of expressing the position of a point in a plane using a distance from a reference point and an angle from a reference direction. This system is different from the more familiar rectangular coordinate system, which measures a point's position using horizontal and vertical distances (x and y).
- Polar coordinates use two values:
- \( r \): the distance from the origin (the center point, often represented as the pole)
- \( \theta \): the angle measured from the positive x-axis (typically in radians)
- The conversion between polar and rectangular coordinates relates through the equations:
- \( x = r\cos\theta \)
- \( y = r\sin\theta \)
- \( r = \sqrt{x^2 + y^2} \)
- \( \theta = \tan^{-1}\left(\frac{y}{x}\right) \)
Rectangular Coordinates
Rectangular coordinates, also known as Cartesian coordinates, are a system where each point in a plane is determined by an ordered pair of numbers, typically referred to as \( (x, y) \). This system is based on a grid formed by perpendicular horizontal and vertical lines.
- The x-coordinate measures how far a point is to the left or right of the origin (zero point), while the y-coordinate measures how far a point is above or below the origin.
- Rectangular equations can represent a variety of shapes including lines, circles, ellipses, parabolas, and hyperbolas.
- \( x = r\cos\theta \)
- \( y = r\sin\theta \)
Graphing Circles
Graphing circles in the coordinate plane involves understanding the standard form of a circle's equation, which is \[(x - h)^2 + (y - k)^2 = r^2,\]where \((h, k)\) is the circle's center, and \(r\) is its radius.
- This equation shows that all points on the circle are equidistant from the center, a key property defining a circle.
- Graphically, this means moving \(r\) units away from the center in all directions to form a perfect circle.
- \((x - 2)^2 + y^2 = 4\) for the first circle, translating to a circle centered at \((2, 0)\) with radius 2.
- \((x + 2)^2 + y^2 = 4\) for the second circle, meaning a circle centered at \((-2, 0)\) with a radius of 2.
